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The value of demand information in an insurance market under demand and cost uncertainty.

JEL G22 * L13 ? D81

Introduction

Effective decision making requires accurate demand information, which is thus important for competing with rivals. In fact, many firms use market research to survey consumers for demand information about their tastes, actions, potential needs, and so on. In particular, increased demand uncertainty in a market raises the value of demand information and generates greater incentives for firms to acquire that information. Demand information is also critical for firms selling insurance products.

In addition, many firms face cost uncertainty and insurance firms are no exception. For any insurance product, an insurance firm must set an insurance premium, which is equivalent to the price, and the insurance payment, which is equivalent to the cost. Thus, insurance firms face cost uncertainty because they do not know how much of an insurance payment will have to be paid in advance. (1) This paper investigates the value of demand information in an insurance market under both demand and cost uncertainty, something not previously considered in the literature. From that viewpoint, this research seeks to shed light on the characteristics of the value of demand information in an insurance market under demand and cost uncertainty.

Literature Review

Many studies have analyzed demand uncertainty. For example, Basar and Ho (1974) considered a Cournot duopoly model with demand uncertainty and checked the properties of the Nash equilibrium. Levine and Ponssard (1979) investigated the value of demand information using game-theoretical models. Clarke (1983) investigated whether an informed player has an incentive to share his/her information. Vives (1984) analyzed both Cournot and Bertrand competition in a duopoly market under demand uncertainty. Sakai and Yoshizumi (1991) considered a duopoly market in which two players are risk averse.

Likewise, many studies have analyzed cost uncertainty. For example, Sakai (1985) investigated and compared sixteen information structures in which two players have/do not have their own/their rival's cost information. Sakai (1986) considered a cost uncertainty model in a differentiated product market. Gal-Or (1986) discussed the incentives for sharing private cost information. Sakai and Yamato (1990) demonstrated how to affect social welfare when players exchange their own cost information.

The Characteristics and Purpose of this Research

There are several previous studies on cost uncertainty in insurance markets. In a pioneering study, Polbom (1998) investigated an insurance market under cost uncertainty and considered the situation in which risk-averse insurance firms compete on insurance premiums. Although the structure of our model is similar to that of Polbom (1998), there are a number of differences.

First, in our model, insurance firms compete on quantities. This is because capacities restrict the sales volumes of insurance firms. In the case of life insurance, the number of underwriters can be interpreted as the capacity of each life insurance firm. Furthermore, suppose that consumers recognize that all insurance products are identical. In this case, from the results in Kreps and Scheinkman (1983), we know the outcome derived from our model is the same as that derived from a two-stage game incorporating capacity and price competition.

Second, in our model the degree of absolute risk aversion is endogenous rather than (implicitly) exogenous as in Polborn (1998). In our model, insurance firms can change their own degree of absolute risk aversion by increasing their capital investment.

Third, this research investigates the effects of both demand and cost uncertainty and considers the value of demand information in the insurance market. Although Sakai (1993) and Asplund (2002) investigated a market incorporating both demand and cost uncertainty, these papers did not investigate the relationship between the two kinds of uncertainty.

The purpose of this research is to investigate the value of demand information in an insurance market under both demand and cost uncertainty. In particular, the following three questions are considered. The first question is whether the value of demand information is always positive. The second question is how to change the value of demand information when exogenous conditions such as the magnitude of cost uncertainty change. The third question is how to change the value of demand information when the number of insurance firms changes.

The Model

Suppose that there are n[greater than or equal to]2 identical insurance firms in an insurance market and that all insurance firms are weakly risk averse about cost uncertainty. We denote by [x.sup.k.sub.i][greater than or equal to]0 the amount of insurance payment paid to k, who is insured by insurance firm i, where i [member of] {1, 2, ... n}. No insurance firms know the exact amount of insurance payments in advance. They only know the form of the probability distribution function for the amount of insurance payment. For simplicity, we assume that each [x.sup.k.sub.i] is mutually independent and that they all have the same mean and variance. In addition, for tractability of the analysis, [x.sup.k.sub.i] is normally distributed as N([[mu].sub.x], [[sigma].sup.2.sub.x]), where [[mu].sub.x] [equivalent to] E[[x.sup.k.sub.i]], [[sigma].sup.2.sub.x] [equivalent to] E[[([x.sup.k.sub.i] - [[mu].sub.x]).sup.2]], and E[*] represents the expectations operator. In our model, [[sigma].sup.2.sub.x] represents the magnitude of cost uncertainty in an insurance market. All insurance firms take part in a two-stage game. They simultaneously choose their amount of investment in the first stage and choose their quantities in the second stage.

First Stage

Investing can strengthen a firm's capital condition and reduce its degree of absolute risk aversion. The investment cost function is

[C.sub.i] = 1/2 [cr.sup.2.sub.i], (1)

where [r.sub.i] represents not only the investment level but also the reduction in insurance firm i's degree of absolute risk aversion induced by the investment. Thus, an increase in [r.sub.i] means not only increasing the investment level and cost but also decreasing the degree of absolute risk aversion and the risk premium. c>0 denotes the constant value of the investment cost function. This parameter, called the "investment parameter" in this research, represents an insurance firm's capability to strengthen its capital. It consists of the magnitude of the marginal investment cost.

The cost function presented in Eq. 1 implies that extra spending on [r.sub.i] has diminishing returns, so a unit reduction in the degree of absolute risk aversion requires an increasing amount of spending. In addition, Eq. 1 means that if an insurance firm does not invest at all--that is, if [r.sub.i]=0--the investment cost is also zero; that is, [C.sub.i]=0.

Second Stage

In the second stage, after observing the amount of investment of all insurance firms, each insurance firm simultaneously chooses its quantity, which is denoted by [q.sub.i].

The profit of each insurance firm, which is denoted by [[PI].sub.i], is

[[PI].sub.i] = [pq.sub.i] - [[q.sub.i].summation over (k=1)] [x.sup.k.sub.i] - 1/2 [cr.sup.2.sub.i], (2)

where p>0 represents the insurance premium. The demand function in the insurance market is assumed to be linear and is specified as

p = a - [n.summation over (i=1)] [q.sub.i], (3)

where a>0 is the intercept of the demand function, which represents the insurance market's maximum (potential) demand. (2)

In our model, if all insurance firms have demand information, they know a and can choose their levels of investment and quantity based on Eq. 3. In contrast, if all insurance firms do not have demand information, they only know the form of the probability distribution function for a. For simplicity, assume that a is normally distributed as N([[mu].sub.a], [[sigma].sup.2.sub.a]), where [[mu].sub.a][equivalent to]E[a] and [[sigma].sup.2.sub.a] [equivalent to] E[[(a - [[mu].sub.a]).sup.2]]. In our model, [[sigma].sup.2.sub.a] represents the magnitude of demand uncertainty in an insurance market. Thus, when no insurance firms have demand information, the expected demand function is

p = [[mu].sub.a] - [n.summation over (i=1)] [q.sub.i]. (4)

The utility function of each insurance firm is

[u.sub.i] = -exp(-([bar.r] - [r.sub.i])[[PI].sub.i]), (5)

where [bar.r][greater than or equal to][r.sub.i] represents the initial degree of absolute risk aversion. It represents the degree of risk aversion before investment. In other words, [bar.r] represents the degree of absolute risk aversion when the amount of investment is zero. Thus, if [bar.r] is high, insurance firms are highly risk averse before investment. In addition, the form of the utility function in Eq. 5 is a well-known function in which the degree of absolute risk aversion is independent of the amount of wealth.

Deriving the Certainty Equivalent

From Eqs. 2 and 5, the certainty equivalent of each insurance firm, which is indicated by [CE.sub.i], can be derived (3):

[CE.sub.i] = E[[[PI].sub.i]] - ([bar.r] - [r.sub.i])Var[[[PI].sub.i]]/2, (6)

where Var[*] is the variance operator. According to Seog (2010: 20), "the certainty equivalent (CE) of a gamble is the sure wealth level that provides the individual with the same utility as the expected utility with the gamble." Thus, the certainty equivalent is another kind of representation of utility under uncertainty. Thus, we know that maximizing the certainty equivalent is equivalent to maximizing expected utility.

The expected profit of each insurance firm, which is denoted by E[[[PI].sub.i]] in Eq. 6, is

E[[[PI].sub.i]] = (p - [[mu].sub.x])[q.sub.i] - 1/2 [cr.sup.2.sub.i]. (7)

The variance of the each insurance firm's profit, which is denoted by Var[[[PI].sub.i]] in Eq. 6, is

Var[[[PI].sub.i]] - [[sigma].sup.2.sub.x][q.sub.i]. (8)

Substituting Eqs. 7 and 8 into Eq. 6 yields the certainty equivalent of each insurance firm:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (9)

To ensure that selling insurance generates a nonnegative profit, it is assumed that the following conditions are satisfied:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (10)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (11)

Equilibrium Concept

Choices in investment and quantity are oligopolistic because the decisions of one insurance firm relate to its rivals' certainty equivalents. In addition, the decisions in the second stage are made after observing the decisions in the first stage. Thus, this is a "dynamic game of complete information." (4) Although the Nash equilibrium is a well-known equilibrium concept in oligopolistic markets, the Nash equilibrium is not an appropriate equilibrium concept in the dynamic game because multiple Nash equilibria may appear, and some of them may not be rational in the dynamic game. Thus, other equilibrium concepts, which can remove irrational Nash equilibria, must be introduced. One refined equilibrium concept is the "subgame-perfect equilibrium."

The subgame-perfect equilibrium can be derived by backward induction as it can remove irrational Nash equilibria. Backward induction implies that we begin computing in the final stage and solve backward (Shy 1995). Thus, in our model, we compute the second stage before analyzing the first stage.

The Value of Demand Information

From the analysis of this model (see Appendix A), Proposition 1 is derived.

Proposition 1: The value of demand information is always positive. (5)

This proposition means that obtaining demand information is meaningful for all insurance firms. In other words, insurance firms have an incentive to acquire demand information if there is no cost. This result is intuitive and can explain why (insurance) firms collect demand information.

Changes in Exogenous Conditions

The following proposition shows how the value of demand information changes when demand uncertainty, the investment parameter, and cost uncertainty change.

Proposition 2: Increases in demand uncertainty and the investment parameter raise the value of demand information. In contrast, an increase in cost uncertainty lowers the value of demand information. (6)

Proposition 2 has the following implications: First, an increase in demand uncertainty increases the value of demand information. This result is not surprising because demand information is more useful when demand uncertainty is greater.

Second, an increase in the investment parameter increases the value of demand information. If the investment parameter is high--for example, because of low investment efficiency or high borrowing costs--insurance firms are reluctant to increase investment, and demand information becomes highly valuable.

Third, an increase in cost uncertainty lowers the value of demand information. This result implies that the value of demand information differs among markets (such as those for car insurance, home insurance, life insurance, and so on) because each market has a different level of cost uncertainty. For example, because the life insurance market is probably associated with less cost uncertainty than, for example, the earthquake insurance market, it could be argued that demand information is more valuable in the life insurance market than in the earthquake insurance market.

Fourth, the value of demand information is high when demand uncertainty is high and cost uncertainty is low. Hence, these two types of uncertainty have opposite effects on the value of demand information in an insurance market.

Change in the Number of Insurance Firms

The following proposition shows how the value of demand information changes when the number of insurance firms changes.

Proposition 3: Whether an increase in the number of insurance firms raises the value of demand information depends on the magnitude of cost uncertainty. When cost uncertainty is very low (high), an increase in the number of insurance firms lowers (raises) the value of demand information. (7)

Proposition 3 shows that how the value of demand information changes when the number of insurance firms changes is ambiguous. The reason for this ambiguous result is as follows. The equilibrium quantity always decreases when the number of insurance firms increases. However, whether the equilibrium insurance premium decreases when the number of insurance firms increases is ambiguous. If cost uncertainty is very low, an increase in the number of insurance firms lowers the equilibrium insurance premium. In contrast, if cost uncertainty is very high, an increase in the number of insurance firms raises the equilibrium insurance premium.8 The level of the equilibrium insurance premium is positively related to the degree of demand uncertainty because the degree of demand uncertainty is determined according to the product of the insurance premium and quantity. Thus, when cost uncertainty is very low (high), an increase in the number of insurance firms finally lowers (raises) the value of demand information through lowering (raising) the equilibrium insurance premium.

Concluding Remarks

This paper investigated the value of demand information in an insurance market under both demand and cost uncertainty, something not previously considered in the literature. Our main results, which are summarized in three propositions, are as follows. First, the value of demand information is always positive in an insurance market under both demand and cost uncertainty. This result shows that the value of demand information is also positive even if both demand and cost uncertainty coexist. Second, increases in demand uncertainty and the investment parameter and a decrease in cost uncertainty raises the value of demand information. From this result, we find that demand and cost uncertainty have opposite effects on the value of demand information. Third, whether an increase in the number of insurance firms raises the value of demand information depends on the magnitude of cost uncertainty. When cost uncertainty is very low (high), an increase in the number of insurance firms lowers (raises) the value of demand information.

The results of our research can be applied to other markets in which both demand and cost uncertainty exist. For example, airlines face both kinds of uncertainty because they cannot know the exact fuel prices when they sell airline tickets. Banks face both kinds of uncertainty because they cannot know the extent to which bad loans, which are equal to costs for banks, will occur at the issue date. Thus, for example, we expect that an increase in the number of airlines or banks raises the value of demand information when uncertainty in the fuel market and the lending market, respectively, is very high.

Furthermore, there are several possibilities for extending our model. For example, our model assumes that all insurance firms are identical in terms of, for example, their average insurance payment, their investment cost functions, and their initial degree of absolute risk aversion. Therefore, our model could be extended by incorporating asymmetries. In particular, asymmetric demand information could be incorporated by allowing only one insurance firm to have exact knowledge of demand information. In this case, it would be interesting to investigate whether the informed insurance firm would have an incentive to share its information in an insurance market under demand and cost uncertainty.

DOI 10.1007/sl 1293-014-9433-3

Appendix A

Consider the case in which all insurance firms have demand information. In the second stage, each insurance firm chooses its own quantity to satisfy the following first-order condition (9):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (A.1)

Because all insurance firms are identical, [q.sup.*.sub.1] = [q.sup.*.sub.2] = ... = [q.sup.*.sub.n] is realized. From Eq. A.1, the equilibrium quantity can be derived as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (A.2)

where an asterisk denotes an equilibrium value. Substituting Eq. A.2 into Eq. 9 yields

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (A.3)

In the first stage, each insurance firm chooses its investment level to satisfy the following first-order condition:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (A.4)

In order to satisfy the second-order condition, the following condition is assumed to be satisfied (10):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (A.5)

Because all insurance firms are identical, [r.sup.*.sub.1] = [r.sup.*.sub.2] = ... = [r.sup.*.sub.n] is realized. From Eq. A.4, the equilibrium amount of investment is derived as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (A.6)

In order to confirm that [r.sup.*.sub.i] is always positive, the following lemma must be checked.

Lemma 1: 4c[(n+1).sup.2] + {(n - 3)n - 2} [[sigma].sup.4.sub.x] > 0 is always satisfied.

Proof: Let f{n) [equivalent to] 4c[(n+1).sup.2] + {(n-3)n-2} [[sigma].sup.4.sub.x] x f(n) is a monotone increasing function of n. Thus, we confirm that f(n) becomes strictly positive in the case of n=2. From Eq. A.5, we find that [[sigma].sup.2.sub.x] < [square root of 9c/2], and thus, f(2)=4(9c-[[sigma].sup.2.sub.x])>0 is proved. Q. E. D.

In order to guarantee that [bar.r][greater than or equal to][r.sub.i], it is assumed that [bar.r] satisfies the following condition: (11)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (A.7)

Combining Eqs. 10 and A.7 yields

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (A.8)

By substituting Eq. A.6 into Eqs. 3, A.2, and A.3, the following equilibrium values of the quantity, insurance premium, and certainty equivalent can be derived respectively as follows (12):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (A.9)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (A.10)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (A.11)

Appendix B

To determine the value of demand information in an insurance market, we compare the case in which insurance firms have demand information with the case in which they do not. Having discussed the former case in Appendix A, we now consider the other case. When no insurance firms have demand information, the demand function is given by Eq. 4 rather than Eq. 3. Although these two demand functions differ, the same procedure is used to derive the equilibrium. Thus, under the case in which insurance firms do not have demand information, the equilibrium certainty equivalent, which is denoted by [CE.sup.0*.sub.i], is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (B.1)

To show how the certainty equivalent differs depending on whether there is demand information or not, the following equation is derived by Eqs. A.11 and B.1.

[CE.sup.*.sub.i] = [CE.sup.0*.sub.i] + [DELTA], (B.2)

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (B.3)

Equation B.2 can be computed as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (B.4)

The second term on the right-hand side of Eq. B.4 can be interpreted as the value of demand information because it represents the difference between certainty equivalents in the presence and absence of demand information. To simplify the expression of that differential, the second term on the right-hand side of Eq. B.4 is defined as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (B.5)

In order to confirm that the value of demand information is always positive, [OMEGA]>0 must be confirmed. From Lemma 1, the denominator in Eq. B.5 is always positive. The following lemma is the result in the sign of the numerator in Eq. B.5.

Lemma 2: 8[c.sup.2][(n + 1).sup.4] + c[(n + 1).sup.2]{4 + {n - 12)n}[[sigma].sup.4.sub.x] - n[(n - 2).sup.2][[sigma].sup.8.sub.x]>0.

Proof: Let g{[[sigma].sup.2.sub.x][equivalent to]c[(n + 1).sup.2] {4 + {n-2)n}[[sigma].sup.4.sub.x] - n[(n - 2).sup.2][[sigma].sup.8.sub.x]. In order to prove the above lemma, we check g([[sigma].sup.2.sub.x])> - 8[c.sup.2][(n+1).sup.4] for any [[sigma].sup.2.sub.x]. Also suppose that [lambda][equivalent to][[sigma].sup.4.sub.x]; then, g([lambda]) = c[(n + 1).sup.2]{4 + {n - 12)n} [lambda] - n[(n - 2).sup.2][[lambda].sup.2] x g([lambda]) is the quadratic concave function of [lambda] and has one maximum value.

In order to check g([lambda])> - 8[c.sup.2][(n + 1).sup.4] for any [lambda], two corner values, [lambda] = 0 ([[sigma].sup.2.sub.x]=0) and [lambda] = c[(n + 1).sup.2]/n ([[sigma].sup.2.sub.x] = [square root of c[(n + 1).sup.2]]/n), must be investigated. We find that g(0)=0 and g(c[(n + 1).sup.2]/n)=-8[c.sup.2][(n + 1).sup.4]. Thus, we know that -8[c.sup.2][(n+1).sup.4] is the minimum value. Because [lambda]<c[(n + 1).sup.2]/n ([[sigma].sup.2.sub.x] < [square root of c[(n + 1).sup.2]]/n) must be satisfied by Eq. A.5, g([lambda]) > -8[c.sup.2][(n + 1).sup.4] is always satisfied. Q. E. D. From Lemma 2, we find that the numerator in Eq. B.5 is always positive and that [OMEGA]>0 is confirmed.

Appendix C

In order to determine how the value of demand information changes when the exogenous variables change, the following comparative statics must be calculated:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (C.1)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (C.2)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (C.3)

From Lemma 2 in Appendix B and n(24 + n(n - 9)) - 4 > 0, we find that [partial derivative][OMEGA]/[partial derivative][[sigma].sup.2.sub.a] > 0, [partial derivative][OMEGA]/ [partial derivative] c > 0, and [partial derivative][OMEGA]/[[partial derivative].sup.2.sub.x] <0. (13)

Appendix D

In order to determine how the value of demand information changes when the number of insurance firms changes, comparative statics are calculated as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (D.1)

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (D.2)

Consider the two extreme cases [[sigma].sup.2.sub.x] = 0 ([[sigma].sup.2.sub.x] is very low) and [[sigma].sup.2.sub.x] [approximately equal to] [square root of (c[(n + 1))].sup.2]/n ([[sigma].sup.2.sub.x] is very high). (14) In the first case, [partial derivative][OMEGA]/[partial derivative] n < 0 is realized because [XI] = 64[c.sup.3][(n + 1).sup.6]>0. In the second case, [partial derivative][OMEGA]/[partial derivative]n > 0 is realized because [XI][approximately equal to] - [c.sup.3][(n-1).sup.2][(n - 1).sup.6][(n + 2).sup.3]/[n.sup.3]<0. (15)

Appendix E

By differentiating Eq. A.9 with respect to n and evaluating [[sigma].sup.2.sub.x] = 0 and [[sigma].sup.2.sub.x] = [square root of (c[(n + 1).sup.2])]/n,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (E.1)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (E.2)

In contrast, by differentiating Eq. A. 10 with respect to n and evaluating at [[sigma].sup.2.sub.x] = 0, we show that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (E.3)

In the case of [[sigma].sup.2.sub.x] = [square root of (c[(n + 1).sup.2])]/n , we derive the following:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (E.4)

In order to confirm that the sign of Eq. E.4 is positive, it is sufficient to confirm the case of the maximum [bar.r]. Substituting [bar.r] = 2(a-[[mu].sub.x])/[[sigma].sup.2.sub.x] into Eq. E.4, we have (16)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (E.5)

Acknowledgments The author wishes to thank the editor and anonymous reviewer for their comments. The author would like to acknowledge the financial support by the Japanese Ministry of Education, Culture, Sports, Science and Technology in the form of a Grant-in-Aid for Young Scientists (B), 24730362.

Published online: 1 October 2014

References

Asplund, M. (2002). Risk-averse firms in oligopoly. International Journal of Industrial Organization, 20, 995-1012.

Basar, T., and Ho, Y. (1974). Informational properties of the Nash solutions of two stochastic nonzero-sum games. Journal of Economic Theory, 7, 370-387.

Black, Jr., K., Skipper, H. D., and Black in, K. (2013). Life insurance (14th edition). Lucretian, LLC

Clarke, R. N. (1983). Duopolists don't wish to share information. Economics Letters, 11, 33-36.

Dorfman, M. S. (2008). Introduction to risk management and insurance (9th ed.). New Jersey: Pearson Prentice Hall.

Gal-Or, E. (1986). Information transmission--Cournot and Bertrand equilibria. Review of Economic Studies, 53(1), 85-92.

Kreps, D., and Scheinkman, J. (1983). Quantity pre-commitment and Bertrand competition yield Cournot outcomes. Bell Journal of Economics, 14, 326-337.

Larue, B., and Yapo, V. (2000). Asymmetries in risk and in risk attitude: The duopoly case. Journal of Economics and Business, 52, 435-153.

Levine, P., and Ponssard, J.-P. (1979). The value of information in some nonzero sum games. International Journal of Game Theory, 6, 221-229.

Polborn, M. K. (1998). A model of an oligopoly in an insurance market. Geneva Papers on Risk and Insurance Theory, 23, 41-48.

Sakai, Y. (1985). The value of information in a simple duopoly model. Journal of Economic Theory, 36(1), 36-54.

Sakai, Y. (1986). Cournot and Bertrand equilibria under imperfect information. Journal of Economics, 46(3) 213-232.

Sakai, Y. (1993). The role of information in profit-maximizing and labor-managed duopoly models. Managerial and Decision Economics, 14(5), 419-432.

Sakai, Y., & Yamato, T. (1990). On the exchange of cost information in a Bertrand-type duopoly model. Economic Studies Quarterly, 41(1), 48-64.

Sakai, Y., and Yoshizumi, A. (1991). The impact of risk aversion on information transmission between firms. Journal of Economics, 53, 51-73.

Seog, H. S. (2010). The economics of risk and insurance. West Sussex: Wiley-Blackwell

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M. Okura ([mail])

Faculty of Economics, Nagasaki University, 4-2-1, Katafuchi, Nagasaki 850-8506, Japan e-mail: okura@nagasaki-u.ac.jp

(1) As explained in many insurance texts, cost uncertainty can be reduced by the law of large numbers. However, cost uncertainty remains in practice. Evidence is that insurance firms include "reserves for unexpected losses (unforeseen contingencies)" in insurance premiums. For details, see Dorfman (2008) and Black et al. (2013).

(2) For tractability of the analysis, linear demand functions have been used in many previous studies, including Basar and Ho (1974), Vives (1984), Sakai (1985, 1986), Gar-Or (1986), Sakai and Yamato (1990), and Sakai and Yoshizumi (1991).

(3) This calculation was also used by Polborn (1998) and Larue and Yapo (2000).

(4) For example, Gibbons (1992) covers how to solve such dynamic games of complete information. The Stackelberg model is one of the well-known models categorized by this game.

(5) Proof provided in Appendix B.

(6) Proof provided in Appendix C.

(7) Proof provided in Appendix D.

(8) Proof provided in Appendix E.

(9) The second-order condition is always satisfied.

(10) When Eq. A.5 is not satisfied, [r.sup.*.sub.i] = 0.

(11) [bar.r] in Eq. A.7 always exists because

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(12) [q.sub.i.sup.*] and [p.sup.*][greater than or equal to]0 are always satisfied because we can prove that 2c[(n+1).sup.2]-(n-2)[[sigma].sup.4.sub.x]>0 is always satisfied as follows. 2c[(n+1).sup.2]-(n-2)[[sigma].sup.4.sub.x]>0 can be transformed as [[sigma].sup.4.sub.x]<2c[(n+1).sup.2]/(n- 2). Equation A.5 can be transformed as [[sigma].sup.4.sub.x]<c[(n+1).sup.2]/n. Because 2c[(n+1).sup.2]/(n-2)>c[(n+1).sup.2]/(n-2), 2c[(n+1).sup.2]-(n- 2)[[sigma].sup.4.sub.x]>0 is always satisfied.

(13) h(n)[equivalent to]n(24+n(n-9))-4>0 is easily proved because h(n) becomes the minimum value in the case of n=4, and then, h(4) = 12 > 0.

(14) This maximum value of [[sigma].sup.2.sub.x] comes from Eq. A.5.

(15) However, we cannot know how the magnitude of [[sigma].sup.2.sub.x] globally affects the value of demand information because [XI] is not a monotone function of [sigma].sup.2.sub.x].

(16) The maximum [bar.r] can be found in Eq. A.8.
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